Question

Consider the function f(x) = 1/x on the interval [1, 10] Find the average or mean slope of the function on this interval.

Answer 1

We will have the following:

First, we determine the equation for the slope, that is:

$f(x)=\frac{1}{x}\Rightarrow f^{\prime}(x)=-\frac{1}{x^2}$

Now, to determine the average slope we examine the slopes at the edges:

$\begin{gathered} f^{\prime}(1)=-\frac{1}{1^2}\Rightarrow f^{\prime}(1)=-1 \\ \\ and \\ \\ f^{\prime}(10)=-\frac{1}{(10)^2}\Rightarrow f^{\prime}(10)=-\frac{1}{100} \end{gathered}$

So, the average slope will be given by:

$\begin{gathered} m_a=\frac{(-\frac{1}{100})+(-1)}{2}\Rightarrow m_a=-\frac{101}{200} \\ \\ \Rightarrow m_a=-0.505 \end{gathered}$

So, the average slope in that interval is -0.505.

Now, we determine the exact point where the slope is that, that is:

$\begin{gathered} -0.505=-\frac{1}{x^2}\Rightarrow x^2=\frac{1}{0.505} \\ \\ \Rightarrow x=\sqrt{\frac{1}{0.505}}\Rightarrow x=1.407195089... \\ \\ \Rightarrow x\approx1.4 \end{gathered}$

So, the exact point is sqrt( 1 / 0.505), that is approximately 1.4.